Decremental kurtosis calculation for big data or streamed data using components

ABSTRACT

The present invention extends to methods, systems, and computing system program products for decrementally calculating kurtosis for Big Data or streamed data. Embodiments of the invention include decrementally calculating one or more components of a kurtosis for an adjusted computation subset based on one or more components of the kurtosis calculated for a previous computation subset and then calculating the kurtosis based on the components. Decrementally calculating kurtosis avoids visiting all data elements in the adjusted computation subset and performing redundant computations thereby increasing calculation efficiency, saving computing resources and reducing computing system&#39;s power consumption.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of provisional patent application Ser. No. 62/089,238, filed 2014 Dec. 9 by the present inventor.

BACKGROUND AND RELEVANT ART

Internet, mobile communications, navigation, online gaming, sensing technologies and large scale computing infrastructures are producing large amounts of data every day. Big Data is data that is beyond the processing capacity of conventional database systems and analyzing capacity of traditional analyzing methods due to its large volume and fast moving and growing speed. More companies now rely on Big Data to make real-time decisions to solve various problems. Current methods involve utilizing a lot of computational resources, which are very costly, yet still may not satisfy the needs of real-time decision making based on the newest information, especially in the financial industry. How to efficiently, promptly and cost-effectively process and analyze Big Data presents a difficult challenge to data analysts and computer scientists.

Streamed data is data that is constantly being received by a receiver while being delivered by a provider. Streamed data may be real-time data gathered from sensors and continuously transferred to computing devices or electronic devices. Often this includes receiving similarly formatted data elements in succession separated by some time interval. Streamed data may also be data continuously read from storage devices, e.g., storage devices on multi-computing devices which store a Big Data set. Stream processing has become a focused research area recently due to the following reasons. One reason is that the input data are coming too fast to store entirely for batch processing, so some analysis have to be performed when the data streams in. The second reason is that immediate responses to any changes of the data are required in some application domains, e.g., mobile related applications, online gaming, navigation, real-time stock analysis and automated trading, etc. The third reason is that some applications or electronic devices require streaming processing due to their nature, e.g., audio, video and digital TV, etc.

Processing Big Data or streamed data may include performing calculations on a multiple data buffer so that some number of data elements may be stored. Processing the streamed data elements may include accessing data elements stored in the buffer. When performing a statistical calculation on Big Data or streamed data elements, buffer requirements may be quite large. For example, when calculating a kurtosis a (potentially large) number of data elements may need to be accessed.

Further, some statistical calculations are recalculated as a Big Data set is changed or some outdated data elements are removed. Thus, the (potentially large) number of data elements may be repeatedly accessed. For example, it may be that a kurtosis is calculated for a computation subset whose size n keeps decreasing to exclude the old data element. As such, every time a data element is received, the data element is removed from the computation subset. The n−1 data elements in the modified computation subset are then accessed to recalculate the kurtosis.

When performing a kurtosis calculation on n−1 data elements in traditional ways, all the n−1 data elements in the computation subset will be visited and used.

In addition, algorithms on streamed data processing may be extended to Big Data processing, because Big Data sets are accumulated over time and they may be considered as a data stream with irregular time intervals. Depending on necessity, the computation subset size n could be extremely large, so the data elements in a computation subset could be distributed over a cloud comprising hundreds of thousands of computing devices. Re-performing a kurtosis calculation on Big Data sets after some data changing in traditional ways results in slow response and significant waste of computing resources.

BRIEF SUMMARY

The present disclosure describes methods, systems, and computing system program products for decrementally calculating kurtosis for Big Data or streamed data by decrementally calculating one or more (p (p≥1)) components of a kurtosis and then calculating the kurtosis using one or more decrementally calculated components. Decrementally calculating kurtosis not only avoids visiting all the data elements in the computation subset but also avoids storing the whole computation subset as traditional algorithms would do, and thus improves the calculation efficiency, saves computing resources and reduces computing system's power consumption.

A computing system comprises one or more computing devices and one or more storage media. The computing system decrementally calculates one or more components of a kurtosis starting from an existing non-empty computation subset of a data set on the one or more storage media or a data stream, where the computation subset size counter and one or more components of a kurtosis may have already been initialized. If not, the computation subset size counter and one or more components of a kurtosis may be initialized based on their definitions using the data elements in the computation subset.

The computing system accesses or receives a data element or information about (e.g., index or address of) the data element to be removed from the non-empty computation subset.

The computing system removes the to-be-removed data element from an optional data buffer as needed.

The computing system modifies the computation subset by removing the accessed or received data element from the computation subset and modifies the computation subset size counter by decreasing its value by 1. The non-empty computation subset before the modification becomes a previous computation subset.

The computing system directly decrementally calculates one or more (v (1≤v≤p)) components of a kurtosis for the modified computation subset. Directly decrementally calculating the v components includes directly decrementally calculating each of the v components one by one. Directly decrementally calculating a component includes: accessing the component calculated or initialized for the prior computation subset and removing any contribution of the removed data element from the component mathematically.

The computing system indirectly decrementally calculating w=p−v components as needed: indirectly decrementally calculating the w components includes indirectly decrementally calculating each of the w components one by one. Indirectly decrementally calculating a component includes calculating the component using components other than the component itself.

The computing system generates a kurtosis as needed based on one or more decrementally calculated components.

The computing system may keep accessing or receiving a data element or information about (e.g., index or address of) a data element to be removed from the computation subset, removing the data element from an optional data buffer as needed, modifying the computation subset and the computation subset size, directly decrementally calculating v (1≤v≤p) components, indirectly decrementally calculating w=p−v components as needed and generating a kurtosis as needed, and the computing system may repeat this process for as many times as needed.

This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the Detailed Description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

Additional features and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by the practice of the invention. The features and advantages of the invention may be realized and obtained by means of the instruments and combinations particularly pointed out in the appended claims. These and other features of the present invention will become more fully apparent from the following description and appended claims, or may be learned by the practice of the invention as set forth hereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to describe the manner in which the above-recited and other advantages and features of the invention may be obtained, a more particular description of the invention briefly described above will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. Understanding that these drawings depict only typical embodiments of the invention and are not therefore to be considered to be limiting of its scope, the invention will be described and explained with additional specificity and detail through the use of the accompanying drawings in which:

FIG. 1 illustrates a high-level overview of an example computing system that facilitates decrementally calculating kurtosis for Big Data or streamed data.

FIG. 1A illustrates an example computing system architecture that facilitates decrementally calculating kurtosis for streamed data with all components being directly decrementally calculated.

FIG. 1B illustrates an example computing system architecture that facilitates decrementally calculating kurtosis for streamed data with some components being directly decrementally calculated and some components being indirectly decrementally calculated.

FIG. 1C illustrates an example computing system architecture that facilitates decrementally calculating kurtosis for Big Data with all components being directly decrementally calculated.

FIG. 1D illustrates an example computing system architecture that facilitates decrementally calculating kurtosis for Big Data with some components being directly decrementally calculated and some components being indirectly decrementally calculated.

FIG. 2 illustrates a flow chart of an example method for decrementally calculating a kurtosis for Big Data or streamed data.

FIG. 3A illustrates data element that is accessed from a computation window for decrementally calculating a kurtosis on streamed data.

FIG. 3B illustrates data element that is accessed from a computation subset for decrementally calculating a kurtosis on Big Data.

FIG. 4A illustrates the definition of a kurtosis and traditional equations for calculating a kurtosis.

FIG. 4B illustrates some example components of a kurtosis and basic decremental component calculation equations.

FIG. 4C illustrates the equations of the first example decremental kurtosis calculation algorithm (decremental algorithm 1).

FIG. 4D illustrates the equations of the second example decremental kurtosis calculation algorithm (decremental algorithm 2).

FIG. 4E illustrates the equations of the third example decremental kurtosis calculation algorithm (decremental algorithm 3).

FIG. 5A illustrates an example of calculating kurtosis using traditional algorithms as shown in FIG. 4A.

FIG. 5B illustrates an example of calculating kurtosis using the first example decremental algorithm as shown in FIG. 4C.

FIG. 5C illustrates an example of calculating kurtosis using the second example decremental algorithm as shown in FIG. 4D.

FIG. 5D illustrates an example of calculating kurtosis using the third example decremental algorithm as shown in FIG. 4E.

FIG. 6 illustrates computational loads for traditional kurtosis algorithms and decremental kurtosis algorithms with a computation subset of size 4.

FIG. 7 illustrates computational loads for traditional kurtosis algorithms and decremental kurtosis algorithms with a computation subset of size 1,000,000.

DETAILED DESCRIPTION

The present disclosure describes methods, systems, and computing system program products for decrementally calculating a kurtosis for Big Data or streamed data by decrementally calculating one or more (p (p≥1)) components of a kurtosis. Decrementally calculating kurtosis not only avoids visiting all the data elements in the computation subset but also avoids storing the whole computation subset as traditional algorithms would do, and thus improves the calculation efficiency, saves computing resources and reduces computing system's power consumption. A computing system comprises one or more computing devices and one or more storage media. The computing system maintains a computation subset size counter and one or more components of a kurtosis. The computing system may also optionally include a data buffer for keeping data elements contained in a non-empty computation subset. The computation subset size counter specifies the number of data elements in the non-empty computation subset.

A kurtosis is the normalized fourth central moment of the values of a random variable, which describes how fast the density function approaches zero in the left and right tail. A negative kurtosis indicates that the variance is mostly caused by many values far away from the mean of the values. A positive kurtosis indicates that the variance is mainly affected by few extreme deviations from the mean of the values.

As used herein, a computation subset is a subset of a Big Data set which contains the data elements involved in a kurtosis calculation. A computation subset is equivalent to a moving computation window when performing a kurtosis calculation on streamed data or time series data. Within the description of embodiments of the present invention, the difference between a computation window and a computation subset is that data elements in a computation window are ordered but that in a computation subset are not.

A component of a kurtosis is a quantity or expression appearing in the kurtosis's definition equation or any transforms of the equation. A kurtosis is the single largest component of a kurtosis itself. A kurtosis may be directly calculated using one or more components of kurtosis. Some example components of a kurtosis may be found in FIG. 4B.

A component may be either directly decrementally calculated or indirectly decrementally calculated. The difference between them is that when directly decrementally calculating a component, the component is calculated based on the component's value in previous iteration but when indirectly decrementally calculating a component, the component is calculated based on components other than the component itself.

For a given component, it might be directly decrementally calculated in one algorithm but indirectly decrementally calculated in another algorithm.

For a given algorithm, assume the total number of different components is p (p≥1), the number of directly decrementally calculated components is v (1≤v≤p), then the number of indirectly decrementally calculated components is w=p−v (0≤w<p). For any algorithm, there will be at least one component being directly decrementally calculated. It is possible that all components are directly decrementally calculated (in this case v=p and w=0). However, directly decrementally calculated components must be calculated in every iteration no matter if a kurtosis is accessed or not in a specific iteration.

For a given algorithm, if a component is directly decrementally calculated, then the component must be calculated in every iteration (i.e., whenever a data element is removed from the computation subset). However, if a component is indirectly decrementally calculated, then the component may be calculated as needed using one or more components other than the component itself, i.e., when a kurtosis needs to be calculated and accessed. So, when a kurtosis is not accessed in a specific iteration, only a small number of components are decrementally calculated to save computation time. It should be understood that an indirectly decrementally calculated component may also be used in the calculation of a directly decrementally calculated component. In that case, the indirectly decrementally calculated component should also be calculated in every iteration.

Embodiments of the invention include decrementally calculating each of the one or more components of a kurtosis for an adjusted computation subset based on one or more components calculated for a prior computation subset.

The computing system decrementally calculates one or more components of a kurtosis starting from an existing non-empty computation subset of a data set on one or more storage media or a data stream, where the computation subset size and one or more components of a kurtosis may have already been initialized. If not, the computation subset size and one or more components of a kurtosis may be initialized based on their definitions using the data elements in the computation subset.

The computing system initializes a computation subset size counter n (n>0) and one or more (v (1≤v≤p)) components of a kurtosis for a computation subset of a size specified by the computation subset size counter. The initialization of the computation subset size comprises setting its value with the number of data elements contained in the computation subset or accessing or receiving a specified computation subset size. The initialization of the one or more components comprises calculating the one or more components through their definitions based on the data elements in the computation subset or accessing or receiving pre-calculated one or more components from one or more computing-device-readable media.

For decremental kurtosis calculation algorithms, the input is not a data element to be added to the computation subset but a data element to be removed from the computation subset.

The computing system accesses or receives a data element or information about (e.g., index or address of) the data element to be removed from the non-empty computation subset. The data element to be removed from the computation subset must be one of the existing data elements in the non-empty computation subset, i.e., the data element must have been added to the computation subset earlier.

The computing system removes the to-be-removed data element from a data buffer as needed. Pure decremental kurtosis calculation only needs access to the data element just accessed or received but do not need access to data elements accessed or received earlier than the just accessed or received data element, so the computing system doesn't necessarily allocate a storage space to keep all the data elements of a computation subset but at least always keeps the computation subset size and one or more components of a kurtosis for the computation subset, and modifying a computation subset may be reflected by modifying the computation subset size and one or more components of a kurtosis for the computation subset. However, since iterative kurtosis calculation requires access to earlier accessed or received data elements, when combining decremental kurtosis calculation with iterative kurtosis calculation, the accessed or received data element needs to be removed from a data buffer allocated for iterative kurtosis calculation. Thus, the computing system may optionally have a data buffer for keeping data elements in the computation subset.

The computing system modifies the non-empty computation subset by removing the to-be-removed data element from the non-empty computation subset. The computing system modifies the computation subset size counter by decreasing its value by 1. The non-empty computation subset before the modification becomes a prior computation subset.

The computing system directly decrementally calculates one or more v (1≤v≤p) components of a kurtosis for the modified computation subset. Directly decrementally calculating the v components of a kurtosis includes directly decrementally calculating each of the v components one by one. Directly decrementally calculating a component includes: accessing the component calculated for the prior computation subset and removing any contribution of the removed data element from the component mathematically.

The computing system indirectly decrementally calculating w=p−v components as needed: indirectly decrementally calculating the w components includes indirectly decrementally calculating each of the w components one by one. Indirectly decrementally calculating a component includes calculating the component using components other than the component itself (Depending on a specific algorithm used, calculating each of the w components may also need access and use the data element removed from the computation subset).

The computing system generates a kurtosis as needed based on one or more decrementally calculated components.

The computing system may keep accessing or receiving a data element or information about (e.g., index or address of) the data element to be removed from the non-empty computation subset, removing the to-be-removed data element from an optional data buffer as needed, modifying the computation subset and the computation subset size, directly decrementally calculating v (1≤v≤p) components, indirectly decrementally calculating w=p−v components as needed and generating a kurtosis as needed, and the computing system may repeat this process for as many times as needed.

Embodiments of the present invention may comprise or utilize a special purpose or general-purpose computing device including computing device hardware, such as, for example, one or more processors and system memory, as discussed in greater detail below. Embodiments within the scope of the present invention also include physical and other computing-device-readable media for carrying or storing computing-device-executable instructions and/or data structures. Such computing-device-readable media may be any available media that may be accessed by a general purpose or special purpose computing device. Computing-device-readable media that store computing-device-executable instructions are computing device storage media (devices). Computing-device-readable media that carry computing-device-executable instructions are transmission media. Thus, by way of example, and not limitation, embodiments of the invention may comprise at least two distinctly different kinds of computing-device-readable media: computing device storage media (devices) and transmission media.

Computing device storage media (devices) includes RAM, ROM, EEPROM, CD-ROM, solid state drives (“SSDs”) (e.g., based on RAM), Flash memory, phase-change memory (“PCM”), other types of memory, other optical disk storage, magnetic disk storage or other magnetic storage devices, or any other medium which may be used to store desired program code means in the form of computing-device-executable instructions or data structures and which may be accessed by a general purpose or special purpose computing device.

A “network” is defined as one or more data links that enable the transport of electronic data between computing devices and/or modules and/or other electronic devices. When information is transferred or provided over a network or another communications connection (either hardwired, wireless, or a combination of hardwired or wireless) to a computing device, the computing device properly views the connection as a transmission medium. Transmissions media may include a network and/or data links which may be used to carry desired program code means in the form of computing-device-executable instructions or data structures and which may be accessed by a general purpose or special purpose computing device. Combinations of the above should also be included within the scope of computing-device-readable media.

Further, upon reaching various computing device components, program code means in the form of computing-device-executable instructions or data structures may be transferred automatically from transmission media to computing device storage media (devices) (or vice versa). For example, computing-device-executable instructions or data structures received over a network or data link may be buffered in RAM within a network interface module (e.g., a “NIC”), and then eventually transferred to computing device RAM and/or to less volatile computing device storage media (devices) at a computing device. Thus, it should be understood that computing device storage media (devices) may be included in computing device components that also (or even primarily) utilize transmission media.

Computing-device-executable instructions comprise, for example, instructions and data which, when executed at a processor, cause a general purpose computing device or special purpose computing device to perform a certain function or group of functions. The computing device executable instructions may be, for example, binaries, intermediate format instructions such as assembly language, or even source code. Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the described features or acts described above. Rather, the described features and acts are disclosed as example forms of implementing the claims.

Those skilled in the art will appreciate that embodiments of the present invention may be practiced in network computing environments with many types of computing device configurations, including, personal computers, desktop computers, laptop computers, message processors, hand-held devices, multi-processor systems, microprocessor-based or programmable consumer electronics, network PCs, minicomputers, mainframe computers, supercomputers, mobile telephones, PDAs, tablets, pagers, routers, switches, and the like. Embodiments of the present invention may also be practiced in distributed system environments where local and remote computing devices, which are linked (either by hardwired data links, wireless data links, or by a combination of hardwired and wireless data links) through a network, both perform tasks. In a distributed system environment, program modules may be located in both local and remote memory storage devices.

Embodiments of the invention may also be implemented in cloud computing environments. In this description and the following claims, “cloud computing” is defined as a model for enabling on-demand network access to a shared pool of configurable computing resources. For example, cloud computing may be employed in the marketplace to offer ubiquitous and convenient on-demand access to the shared pool of configurable computing resources. The shared pool of configurable computing resources may be rapidly provisioned via virtualization and released with low management effort or service provider interaction, and then scaled accordingly.

A cloud computing model may be composed of various characteristics such as, for example, on-demand self-service, broad network access, resource pooling, rapid elasticity, measured service, and so forth. A cloud computing model may also expose various service models, such as, for example, Software as a Service (“SaaS”), Platform as a Service (“PaaS”), and Infrastructure as a Service (“IaaS”). A cloud computing model may also be deployed using different deployment models such as private cloud, community cloud, public cloud, hybrid cloud, and so forth. In this description and in the claims, a “cloud computing environment” is an environment in which cloud computing is employed.

FIG. 1 illustrates a high-level overview of an example computing system 100 that facilitates decrementally calculating kurtosis for Big Data or streamed data. Referring to FIG. 1, computing system 100 comprises multiple devices connected by different networks, such as local network, internet and wireless network, etc. The multiple devices include, for example, a data analysis engine 1007, a storage system 1011, live data stream 1006, and multiple distributed computing devices that may schedule data analysis tasks and/or query data analysis results, such as personal computer 1016, hand-held devices 1017 and desktop computer 1018, etc. Data analysis engine 1007 may comprise one or more processors, e.g., CPU 1009 and CPU 1010, one or more system memory, e.g., system memory 1008, and component calculation modules 191 and 131. Component calculation modules 191 and 131 will be illustrated in more details in other figures. Storage system 1011 may comprise one or more storage media, e.g., storage medium 1012 and storage medium 1014, which may be used for hosting Big Data sets. Data sets on storage system 1011 may be accessed by data analysis engine 1007. In general, data stream 1006 may comprise streamed data from different data sources, for example, stock quotes, audio data, video data, geospatial data, web data, mobile communication data, online gaming data, banking transaction data, sensor data, closed-captioning data, etc. To depict a few, real-time data 1000 may comprise data collected from sensor 1001, stock 1002, web 1003 and bank 1004, etc. in real-time. Data analysis engine 1007 may receive data elements from data stream 1006. Data from different data sources may also be stored in storage system 1011 which may be accessed for Big Data analysis. It should be understood that FIG. 100 is provided to introduce a selection of concepts in a much simplified form, for example, distributed devices 1016 and 1017 may need to go through a firewall to connect data analysis engine 1007, and data accessed or received from data stream 1006 and/or storage system 1011 by data analysis engine 1007 may be filtered by data filters, etc.

FIG. 1A illustrates an example computing system architecture 100A that facilitates decrementally calculating kurtosis for streamed data. FIG. 1A illustrates 1007 and 1006 shown in FIG. 1. Referring to FIG. 1A, computing system architecture 100A includes component calculation module 131. Component calculation module 131 may be connected to (or is part of) a network, such as, for example, a Local Area Network (“LAN”), a Wide Area Network (“WAN”), and even the Internet. Accordingly, component calculation module 131 as well as any other connected computing devices and their components, may send and receive message related data (e.g., Internet Protocol (“IP”) datagrams and other higher layer protocols that utilize IP datagrams, such as, User Datagram Protocol (“UDP”), Real-time Streaming Protocol (“RTSP”), Real-time Transport Protocol (“RTP”), Microsoft® Media Server (“MMS”), Transmission Control Protocol (“TCP”), Hypertext Transfer Protocol (“HTTP”), Simple Mail Transfer Protocol (“SMTP”), etc.) over the network. The output of component calculation module 131 will be used as the input of kurtosis calculation module 191, and kurtosis calculation module 191 will generate kurtosis 192.

In general, data stream 190 may be a sequence of digitally encoded signals (e.g., packets of data or data packets) used to transmit or receive information that is in the process of being transmitted. Data stream 190 may stream data elements to computing system architecture 100A. Data stream 190 may stream stored data or be a live stream. Data stream 190 transmits data elements to be removed from an existing non-empty computation subset.

Decremental kurtosis calculation starts from an existing non-empty computation subset where a computation subset size and one or more components of a kurtosis for the computation subset may have already been initialized or calculated. If not, one or more components of a kurtosis may be initialized by accessing all data elements in the computation subset and calculating the components one by one according to its definition using the all data elements in the computation subset. For example, in computing system architecture 100A, components ranging from Cd₁ 141 and Cd_(v) 145 may have been initialized. As depicted in FIG. 1A, component Cd₁ 141 includes contribution 151 (a contribution from data element 101), contribution 152 (a contribution from data element 102), and contributions 153 (contributions from other data elements). Similarly, component Cd_(v) 145 includes contribution 181 (a contribution from data element 101) and contribution 182 (a contribution from data element 102), and contributions 183 (contributions from other data elements).

Computation subset size counter 118 may be implemented in hardware or software. When it is implemented in hardware, counter 118 may be a small device that may update a computation subset size stored in a location in counter 118. When it is implemented in software, counter 118 may be just a variable stored in RAM, hard drive or any other computing-device-readable storage media, and the content of counter 118 is a computation subset size. Computation subset size counter 118 may be used to keep track the size of a computation subset. Whenever receiving a data element, the computing system adjusts the computation subset by removing the data element from the computation subset and adjusts the computation subset size counter 118 by decreasing its content or value by 1. Computation subset size counter 118 may be accessed or received by component calculation module 131, and it may be set to a specific value by reset module 129 when decremental kurtosis calculation starts working on a non-empty computation subset. Within the description of this disclosure, a computation subset size counter is equivalent to a computation subset size and may be used interchangeably.

For example, as depicted in FIG. 1A, when data element 101 is received, the counter 118 will be decreased by 1. Both the computation subset size counter 118 and data element 101 may be accessed or received by component calculation module 131. The data element 101 must be an existing data element in the computation subset.

Subsequently, data element 102 may be received. When this happens, the computation subset counter 118 will decrease its value by 1. Both the modified computation subset size counter 118 and data element 102 may be accessed or received by component calculation module 131. Similarly, data elements 102 must be an existing data element in the computation subset.

Referring to FIG. 1A, computing system architecture 100A also includes an optional data buffer 125. The optional data buffer 125 is useful when combining decremental kurtosis calculation with iterative kurtosis calculation. As streamed data elements are accessed or received, they may be removed from an optional data buffer 125 as needed (i.e., when combining decremental kurtosis calculation with iterative kurtosis calculation).

Referring to computing system architecture 100A, component calculation module 131 comprises v (v=p≥1) component calculation modules for calculating v components for data elements in a computation subset. The number of components v varies depending on which decremental algorithm is used. As depicted in FIG. 1A, component calculation module 131 comprises component Cd₁ calculation module 161 and component Cd_(v) calculation module 162, and there are v−2 other component calculation modules between them. Calculation module 161 comprises initialization module 132 for initializing component Cd₁ and decremental algorithm 133 for directly decrementally calculating component Cd₁. Calculation module 162 comprises initialization module 138 for initializing component Cd_(v) and decremental algorithm 139 for directly decrementally calculating component Cd_(v). Initialization module 132 is configured to initialize component Cd₁ for data elements in a computation subset and initialization module 138 is configured to initialize component Cd_(v) for data elements in a computation subset. Component Cd₁ 141 is the initial value of component Cd₁. Initialization module 132 may be used for an initialization of component Cd₁ or when kurtosis calculations are reset. Initialization module 132 initializes component Cd₁ 141 to be a specific value composed of contribution 151, contribution 152 and contributions 153 passed in by reset module 129 for a non-empty computation subset. Similarly, initialization module 138 may be used for an initialization of component Cd_(v) or when kurtosis calculations are reset. Component Cd_(v) 145 is the initial value of component Cd_(v). Initialization module 138 initializes component Cd_(v) 145 to be a specific value composed of contribution 181, contribution 182 and contributions 183 passed in by reset module 129 for a non-empty computation subset.

Decremental algorithms are also configured to calculate v components for a set of n data elements in a computation subset. Decremental algorithm 133 accesses or receives a prior component Cd₁ value and a data element to be removed from the computation subset as input. Decremental algorithm 133 calculates a component Cd₁ for the modified computation subset based on the prior component Cd₁ value and the accessed or received data element to be removed from the computation subset. Contribution removal module 133A may remove any contribution of the accessed or received data element from the prior component Cd₁. Removing any contribution of the accessed or received data element may be used for calculating component Cd₁ for the modified computation subset. Decremental algorithm 139 works in a similar way as decremental algorithm 133. Decremental algorithm 139 accesses or receives a prior component Cd_(v) value and a data element to be removed from the computation subset as input. Decremental algorithm 139 calculates a component Cd_(v) for the modified computation subset based on the prior component Cd_(v) value and the accessed or received data element. Contribution removal module 139A may remove any contribution of the accessed or received data element from the prior component Cd_(v). Removing any contribution of the accessed or received data element may be used for calculating component Cd_(v) for the modified computation subset.

Referring to FIG. 1A, computing system architecture 100A also includes kurtosis calculation module 191. Once p (p=v≥1) components of a kurtosis are directly decrementally calculated by component calculation module 131, kurtosis calculation module 191 may calculate the kurtosis 192 using one or more decrementally calculated or initialized components.

FIG. 1B illustrates an example computing system architecture 100B that facilitates decrementally calculating kurtosis for streamed data with some (v (1≤v<p)) components being directly decrementally calculated and some (w=p−v) components being indirectly decrementally calculated. The number v as well as the number w is algorithm dependent. Many parts included in computing system architectures 100B and 100A have same reference numbers. Those parts have similar structures and work in similar ways. In certain implementations, the difference between computing system architectures 100B and 100A may be that architecture 100B includes a component calculation module 135. All parts except component calculation module 135 in 100B work in a similar way as those parts with the same reference numbers in 100A. Instead of repeating what have already been explained in the description about 100A, only the different part is discussed here. Computing system architecture 100B also includes component calculation module 131, which also includes v component calculation modules for directly decrementally calculating v components, however the number v in 100B may not be the same number v as in 100A, because some directly decrementally calculated components in 100A are indirectly decrementally calculated in 100B. In 100A, v=p≥1, but in 100B, 1≤v<p. Referring to FIG. 1B, computing system architecture 100B includes component calculation module 135. The output of components calculation module 131 may be used as the input of component calculation module 135, and the output of calculation modules 131 and 135 may be used as the input of kurtosis calculation module 191, and kurtosis calculation module 191 may generate kurtosis 192. Component calculation module 135 generally includes w=p−v component calculation modules for indirectly decrementally calculating w components. For example, Component calculation module 135 includes calculation module 163 for indirectly decrementally calculating component Ci₁ and calculation module 164 for indirectly decrementally calculating component Ci_(w), and there are w−2 component calculation modules in between. Indirectly decrementally calculating w components includes indirectly decrementally calculating each of the w components one by one. Indirectly decrementally calculating a component includes accessing and using one or more components other than the component itself. The one or more components may have been initialized, directly decrementally calculated or indirectly decrementally calculated.

Referring to computing system architecture 100B, once all p (p=v+w) components have been calculated, kurtosis calculation module 191 may be used for calculating a kurtosis 192 as needed based on one or more decrementally calculated or initialized components.

FIG. 1C illustrates an example computing system architecture 100C that facilitates decrementally calculating kurtosis for streamed data. FIG. 1C illustrates 1007 and 1006 shown in FIG. 1. Referring to FIG. 1C, computing system architecture 100C includes component calculation module 131. Component calculation module 131 may be tightly coupled with one or more storage media by a high-speed data bus or loosely coupled with one or more storage media managed by a storage system via (or is part of) a network, such as, for example, a Local Area Network (“LAN”), a Wide Area Network (“WAN”), and even the Internet.

Storage media 121 comprises a data set 123. As depicted, data set 123 comprises multiple data elements stored in multiple locations of storage media 121. For example, data elements 101, 102, 103, 104, 105, 106 are stored in locations 121A, 121B, 121C, 121D, 121E and 121F respectively, etc., . . . . There are multiple data elements stored in other locations of storage media 121.

Assume that a kurtosis calculation is started from computation subset 122. Computation subset 122 with a specified size of 6 (i.e., n=6) contains a number of data elements of data set 123: data elements 101, 102, 103, 104, 105 and 106. All the data elements ranging from 101 to 106 in computation subset 122 may be accessed for initializing one or more components of the kurtosis.

Next, data element 101 may be accessed from locations 121A. Data element 101 will be removed from computation subset 122, then computation subset 122 becomes modified computation subset 122A. Modified computation subset 122A contains 5 data elements: 102, 103, 104, 105 and 106. The computing system may decrementally calculate a kurtosis for modified computation subset 122A.

Subsequently, data element 102 may be accessed from locations 121B. Data element 102 will be removed from modified computation subset 122A, then modified computation subset 122A becomes modified computation subset 122B. Modified computation subset 122B contains 4 data elements: 103, 104, 105 and 106. The computing system may decrementally calculate a kurtosis for modified computation subset 122B.

Referring to computing system architecture 100C, component calculation module 131 comprises v (v=p≥1) component calculation modules for calculating v components for data elements in a computation subset. The number of components, v, varies depending on which decremental algorithm is used. As depicted in FIG. 1C, component calculation module 131 comprises component Cd₁ calculation module 161 and component Cd_(v) calculation module 162, and there are v−2 other component calculation modules between them. Component Cd₁ calculation module 161 comprises initialization module 132 for initializing component Cd₁ and decremental algorithm 133 for directly decrementally calculating component Cd₁. Component Cd_(v) calculation module 162 comprises initialization module 138 for initializing component Cd_(v) and decremental algorithm 139 for directly decrementally calculating component Cd_(v). Initialization module 132 is configured to initialize component Cd₁ for data elements in a computation subset and initialization module 138 is configured to initialize component Cd_(v) for data elements in a computation subset. Component Cd₁ 141 is the initial value of component Cd₁. Initialization module 132 may be used for an initialization of component Cd₁ or when kurtosis calculations are reset. Initialization module 132 initializes component Cd₁ 141 to be a specific value composed of contribution 151, contribution 152 and contributions 153 passed in by reset module 129 for a non-empty computation subset. Similarly, initialization module 138 may be used for an initialization of component Cd_(v) or when kurtosis calculations are reset. Component Cd_(v) 145 is the initial value of component Cd_(v). Initialization module 138 initializes component Cd_(v) 145 to be a specific value composed of contribution 181, contribution 182 and contributions 183 passed in by reset module 129 for a non-empty computation subset.

Decremental algorithms are also configured to calculate v components for a set of n data elements in a computation subset. Decremental algorithm 133 accesses or receives a prior component Cd₁ value and a data element to be removed from the computation subset as input. Decremental algorithm 133 calculates a component Cd₁ for the modified computation subset based on the prior component Cd₁ value and the data element removed from the computation subset. Contribution removal module 133A may remove any contribution of the removed data element from the prior component Cd₁. Removing any contribution of the removed data element may be used for directly decrementally calculating component Cd₁ for the modified computation subset. Decremental algorithm 139 works in a similar way as decremental algorithm 133. Decremental algorithm 139 receives a prior component Cd_(v) value and a data element to be removed from the computation subset as input. Decremental algorithm 139 calculates a component Cd_(v) for the modified computation subset based the prior component Cd_(v) value and the data element removed from the computation subset. Contribution removal module 139A may remove any contribution of the removed data element from the prior component Cd_(v). Removing any contribution of the removed data element may be used for directly decrementally calculating component Cd_(v) for the modified computation subset.

Referring to FIG. 1C, computing system architecture 100C also includes kurtosis calculation module 191. Once p (p=v≥1) components of a kurtosis are directly decrementally calculated by component calculation module 131, kurtosis calculation module 191 may calculate the kurtosis 192 using one or more decrementally calculated or initialized components.

FIG. 1D illustrates an example computing system architecture 100D that facilitates decrementally calculating kurtosis for Big Data with some (v (1≤v<p)) components being directly decrementally calculated and some (w=p−v) components being indirectly decrementally calculated. The number v as well as the number w is algorithm dependent. Many parts included in computing system architectures 100D and 100C have same reference numbers. Those parts have similar structures and work in similar ways. In certain implementations, the difference between computing system architectures 100D and 100C may be that architecture 100D includes a component calculation module 135. All parts except component calculation module 135 in 100D work in a similar way as those parts with same reference numbers in 100C. Instead of repeating what have already been explained in the description about 100C, only the different part is discussed here. Computing system architecture 100D also includes component calculation module 131, which also includes v component calculation modules for directly decrementally calculating v components, however the number v in 100D may not be the same number v as in 100C, because some directly decrementally calculated components in 100C are indirectly decrementally calculated in 100D. In 100C, v=p≥1, but in 100D, 1≤v<p. Referring to FIG. 1D, computing system architecture 100D includes component calculation module 135. The output of components calculation module 131 may be used as the input of component calculation module 135, and the output of calculation modules 131 and 135 may be used as the input of kurtosis calculation module 191, and kurtosis calculation module 191 may generate kurtosis 192. Component calculation module 135 generally includes w=p−v component calculation modules for indirectly decrementally calculating w components. For example, component calculation module 135 includes component calculation module 163 for indirectly decrementally calculating component Ci₁ and component calculation module 164 for indirectly decrementally calculating component Ci_(w), and there are w−2 component calculation modules in between. Indirectly decrementally calculating w components includes indirectly decrementally calculating each of the w components one by one. Indirectly decrementally calculating a component includes accessing and using one or more components other than the component itself. The one or more components may have been initialized, directly decrementally calculated or indirectly decrementally calculated.

Referring to computing system architecture 100D, once all p (p=v+w) components have been calculated, kurtosis calculation module 191 may be used for calculating a kurtosis 192 as needed based on one or more decrementally calculated or initialized components.

FIG. 2 illustrates a flow chart of an example method 200 for decrementally calculating kurtosis for Big Data or streamed data. Method 200 will be described with respect to the components and data of computing system architectures 100A, 100B, 100C and 100D.

Method 200 includes initializing computation subset size counter 118 and v (1≤v≤p, p≥1) components of a kurtosis for a computation subset of size n (n>1) (201). For example, initialization module 132 may initialize component Cd₁ 141 with contribution 151 (a contribution from data element 101), contribution 152 (a contribution from data element 102), and contributions 153 (contributions from other data elements). Similarly, initialization module 138 may initialize component Cd_(v) 145 with contribution 181 (a contribution from data element 101), contribution 182 (a contribution from data element 102), and contribution 183 (contributions from other data elements).

Method 200 includes accessing or receiving a data element to be removed from the computation subset (202). For example, data element 101 may be accessed or received. Method 200 includes removing the accessed or received data element from an optional data buffer as needed (203). For example, when combining decremental kurtosis calculation with iterative kurtosis calculation, data element 101 may be removed from an optional data buffer 125 as needed. Method 200 includes removing the accessed or received data element from the computation subset and modifying the computation subset size counter (204). For example, data element 101 may be removed from the computation subset and computation subset size counter 118 may be modified by decreasing its current value by 1 upon accessing or receiving the data element 101.

Method 200 includes directly decrementally calculating v (1≤v≤p) components of a kurtosis for the modified computation subset based on v components for the prior computation subset (205). For example, decremental algorithm 133 may be used for directly decrementally calculating component Cd₁ 143 based on component Cd₁ 141 and data element 101, and decremental algorithm 139 may be used for directly decrementally calculating component Cd_(v) 147 based on component Cd_(v) 145 and data element 101.

Directly decrementally calculating v components of a kurtosis for the modified computation subset includes accessing the v components of the kurtosis in the prior computation subset (207). For example, decremental algorithm 133 may access component Cd₁ 141. Similarly, decremental algorithm 139 may access component Cd_(v) 145.

Directly decrementally calculating v components of a kurtosis for the modified computation subset includes removing any contribution of the accessed or received data element from each of the v components mathematically (208). For example, decrementally calculating component Cd₁ 143 may include contribution removal module 133A removing contribution 151 from component Cd₁ 141 mathematically, and decrementally calculating component Cd_(v) 147 may include contribution removal module 139A removing contribution 181 from component Cd_(v) 145 mathematically. Contribution 151 and 181 are contributions of data element 101.

As depicted in FIG. 1A, component Cd₁ 143 includes contribution 152 (a contribution from data element 102) and contributions 153 (contributions from other data elements). Similarly, component Cd_(v) 147 includes contribution 182 (a contribution from data element 102) and contribution 183 (contributions from other data elements).

Besides directly decrementally calculated components, some components may be indirectly decrementally calculated. Method 200 includes indirectly decrementally calculating each of w=p−v components one by one as needed based on one or more components other than the component itself (209) when v<p, i.e., not all components are directly decrementally calculated. Since the w indirectly decrementally calculated components are calculated using other components, they might not need to be initialized. In addition, the w indirectly decrementally calculated components might not need to be calculated in every iteration (i.e., an existing data element removed from the computation subset). They only need to be calculated when a kurtosis is accessed to save computation time. For example, referring to FIG. 1B where some components are directly decrementally calculated and some are indirectly decrementally calculated, calculation module 163 may indirectly decrementally calculate Ci₁ and calculation module 164 may indirectly decrementally calculate Ci_(w). For a given component, it may be directly decrementally calculated in one algorithm but indirectly decrementally calculated in another algorithm. For example, referring to FIG. 4C, in Decremental Algorithm 1, component QX_(k+1) for the k+1^(th) iteration is directly decrementally calculated based on component QX_(k) for the k^(th) iteration and other components and the data element removed from the computation subset. However, referring to FIG. 4E, in Decremental Algorithm 3, component QX_(k+1) for the k+1^(th) iteration is indirectly decrementally calculated based on components other than QX. For a given algorithm, directly decrementally calculated components must be calculated whenever a data element is removed from the computation subset, no matter if a kurtosis is accessed or not; However, an indirectly decrementally component may be calculated as needed, i.e., it only needs to be calculated when a kurtosis is accessed. For example, referring to FIG. 4E, in Decremental Algorithm 3, component QX_(k+1) only needs to be calculated when a kurtosis γ_(k+1) ⁽²⁾ is accessed.

Method 200 includes calculating a kurtosis using one or more decrementally calculated components of the kurtosis (210). The one or more components may all be directly decrementally calculated (v=p) (205) or partially be directly decrementally calculated and partially be indirectly decrementally calculated (v<p) (209).

202-208 may be repeated as more data elements to be removed are accessed or received. 209-210 may be repeated as needed. For example, subsequent to calculating components ranging from Cd₁ 143 to Cd_(v) 147, data element 102 may be accessed or received.

Decremental algorithm 133 may be used for directly decrementally calculating component Cd₁ 144 based on component Cd₁ 143. Decremental algorithm 133 may access component Cd₁ 143. Directly decrementally calculating component Cd₁ 144 may include contribution removal module 133A removing contribution 152 from component Cd₁ 143 mathematically. Contribution 152 is a contribution from data element 102.

Similarly, decremental algorithm 139 may be used for directly decrementally calculating component Cd_(v) 148 based on component Cd_(v) 147. Decremental algorithm 139 may access component Cd_(v) 147. Directly decrementally calculating component Cd_(v) 148 may include contribution removal module 139A removing contribution 182 from component Cd_(v) 147 mathematically. Contribution 182 is a contribution from data element 102.

As depicted, component Cd₁ 144 includes contributions 153 (contributions from other data elements), and component Cd_(v) 148 includes contribution 183 (contributions from other data elements).

When a next data element is accessed or received, component Cd₁ 144 may be used to decrementally calculate a component Cd₁ for the modified computation subset and component Cd_(v) 148 may be used to decrementally calculate a component Cd_(v) modified computation subset.

As depicted, reset 211 may be used for resetting decremental kurtosis calculation. When reset 211 is invoked either after 205 or 210, the computation subset size counter and v components of a kurtosis will be initialized. For example, component Cd₁ 141 may be initialized to be a specific value when the computation subset is non-empty and component Cd₁ 141 has already been calculated. Such case may happen when combining decremental kurtosis calculation with iterative kurtosis calculation or incremental kurtosis calculation. Component Cd_(v) 145 may be initialized in the same way.

FIG. 3A illustrates data that is accessed from a computation window 300A for decrementally calculating a kurtosis on streamed data. Computation window 300A should be non-empty where v components have already been calculated or initialized. The difference between a computation window and a computation subset is that the data elements in a computation window are ordered (e.g., the data element is always removed from either the left end or the right end of a computation window). For example, referring to FIG. 3A, a least recently accessed or received data element is always removed from the left end of computation window 300A. As time progresses, an existing data element, for example, data element x₁, then x₂, then x₃, . . . is removed from the computation window and accessed respectively for directly decrementally calculating next v components of a kurtosis, indirectly decrementally calculating w=p−v components, and calculating the kurtosis using one or more decrementally calculated components. The v components may be directly decrementally calculated using the data element removed from the computation window and the v components for the prior computation window, and other data elements in computation window 300A are not touched. For a given decremental algorithm, v is a constant, so the number of operations for directly decrementally calculating v components is a constant, and the number of operations for indirectly decrementally calculating w=p−v components is also a constant. Thus, the computation workload for calculating all p components for a given computation window with size n is reduced and a kurtosis may be calculated using one or more decrementally calculated components. The larger the n, the more substantial the reduction in computation workload.

FIG. 3B illustrates data that is accessed from a computation subset 300B for decrementally calculating a kurtosis on Big Data. Computation subset 300B should be non-empty where v components have already been calculated or initialized. The difference between a computation subset and a computation window is that the data elements in a computation subset are not ordered (e.g., a data element may be removed from any position of a computation subset). For example, referring to FIG. 3B, an existing data element may be removed from any position of computation subset 300B. As time progresses, an existing data element in computation subset 300B, for example, data element x_(r) at the position indicated by shaded area is removed from the computation subset and accessed for directly decrementally calculating v components of a kurtosis for the modified computation subset, indirectly decrementally calculating w=p−v components, and calculating the kurtosis using one or more decrementally calculated components. The v components may be directly decrementally calculated using the data element removed from the computation subset and the v components for the prior computation subset, and other data elements in computation subset 300B are not touched. For a given decremental algorithm, v is a constant, so the number of operations for directly decrementally calculating v components is a constant, and the number of operations for indirectly decrementally calculating w=p−v components is also a constant. Thus, the computation workload for calculating all p components for a given computation subset with size n is reduced and a kurtosis may be calculated using one or more decrementally calculated components. The larger the n, the more substantial the reduction in computation workload.

FIG. 4A illustrates the definition of kurtosis. Suppose a computation subset X={x₁|i=1, . . . , n} is a subset of a Big Data set or streamed data which contains the data elements to be involved in kurtosis calculation. Equation 401 is a traditional equation for calculating a sum S_(k) of all the data elements in X in the k^(th) iteration. Equation 402 is a traditional equation for calculating a mean x _(k) of all the data elements in X in the k^(th) iteration. Equation 403 is a traditional equation for calculating a kurtosis γ_(k) ⁽²⁾ of all the data elements in a computation subset of size n. Assuming the computation subset X of size n is changed with a data element x_(r) being removed. Define the modified computation subset as X′. The modified computation subset size becomes n−1. The iteration becomes k+1^(th) iteration. Equation 404 is a traditional equation for calculating a sum S_(k+1) of all the data elements in the modified computation subset X′ in the k+1^(th) iteration. Equation 405 is a traditional equation for calculating a mean x _(k+1) of all the data elements in the modified computation subset X′ in the k+1^(th) iteration. Equation 406 is a traditional equation for calculating a kurtosis γ_(k+1) ⁽²⁾ of all the data elements in the modified computation subset X′ in the k+1^(th) iteration.

FIG. 4B illustrates some components of a kurtosis and basic decremental component calculation equations. A component of a kurtosis is a quantity or expression appearing in the kurtosis's definition equation or any transforms of the definition equation. The following are a few example components of the kurtosis. S _(k)=Σ₁ ^(n) x _(i)

${\overset{\_}{x}}_{k} = {\frac{s_{k}}{n} = {\frac{1}{n}{\sum\limits_{1}^{n}x_{i}}}}$ SS _(k)=Σ₁ ^(n) x _(i) ² SX _(k)=Σ₁ ^(n)(x _(i) −x _(k))² CS _(k)=Σ₁ ^(n) x _(i) ³ CX _(k)=Σ₁ ^(n)(x _(i) −x _(k))³ QS _(k)=Σ₁ ^(n) x _(i) ⁴ QX _(k)=Σ₁ ^(n)(x _(i) −x _(k))⁴

$\frac{1}{n}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}}$

$\frac{1}{n}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{3}}$

$\frac{1}{n}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{4}}$

$\left( {\frac{1}{n}{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}}} \right)^{2}$

$\gamma_{k}^{(2)} = \frac{n{\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{4}}}{\left( {\sum\limits_{1}^{n}\left( {x_{i} - {\overset{\_}{x}}_{k}} \right)^{2}} \right)^{2}}$ A kurtosis may be calculated based on one or more components or combinations of them, so there are multiple algorithms supporting decremental kurtosis calculation. To illustrate how to use components to decrementally calculate kurtosis, three different decremental kurtosis calculation algorithms are given as examples. A new iteration of calculation is started each time any component of a kurtosis is recalculated due to a data change in the computation subset which causes a computation subset to change to a modified computation subset. A sum or a mean is the basic component to be used for calculating a kurtosis. Equation 407 is for directly decrementally calculating a sum S_(k+1) of all the data elements in the modified computation subset X′ in the k+1^(th) iteration. Equation 408 is for directly decrementally calculating a mean x _(k+1) of all the data elements in the modified computation subset X′ in the k+1^(th) iteration. Either a sum S_(k+1) or a mean x ₊₁ will be used in all three decremental kurtosis calculation algorithms described later.

FIG. 4C illustrates the first example decremental kurtosis calculation algorithm (decremental algorithm 1). Decremental algorithm 1 comprises decremental calculation of components S_(k+1) or x _(k+1), SS_(k+1), SX_(k+1), CS_(k+1) and QX_(k+1), and a kurtosis γ_(k+1) ⁽²⁾ may then be calculated based on components SX_(k+1) and QX_(k+1) once they are calculated. Equation 407 may be used for directly decrementally calculating component S_(k+1) if component S_(k) is available. Equation 408 may be used for directly decrementally calculating component x _(k+1) if component x _(k) is available. Equation 409 is a traditional equation for calculating component SS_(k) in the computation subset X. Equation 410 is a traditional equation for calculating component SS_(k+1) in the modified computation subset X′. Equation 411 may be used for directly decrementally calculating component SS_(k+1) in the modified computation subset X′ if component SS_(k) is available. Equation 412 is a traditional equation for calculating component SX_(k) in the computation subset X. Equation 413 is a traditional equation for calculating component SX_(k+1) in the modified computation subset X′. Equations 414 may be used for indirectly decrementally calculating component SX_(k+1) in the modified computation subset X′ if components S_(k+1) or x _(k+1) and SS_(k+1) are available. Equations 414 comprise multiple equations but only one of them is needed depending on if a sum S_(k+1) or a mean 4 ₊₁ or both are available. Equation 415 is a traditional equation for calculating component CS_(k) in the computation subset X. Equation 416 is a traditional equation for calculating component CS_(k+1) in the modified computation subset X′. Equation 417 may be used for directly decrementally calculating component CS_(k+1) in the modified computation subset X′ if component CS_(k) is available. Equation 418 is a traditional equation for calculating component QX_(k) in the computation subset X. Equation 419 is a traditional equation for calculating component QX_(k+1) in the modified computation subset X′. Equations 420 may be used for directly decrementally calculating component QX_(k+1) in the modified computation subset X′ if components QX_(k), S_(k), CS_(k+1), SS_(k+1), x _(k) and S_(k+1) or x ₊₁ are available. Equations 420 comprise multiple equations but only one of them is needed depending on if a sum or a mean or both are available. Equation 421 may be used for calculating the kurtosis γ_(k+1) ⁽²⁾ for the modified computation subset X′ based on components QX_(k+1) and SX_(k+1) once they are calculated.

FIG. 4D illustrates the second example decremental kurtosis calculation algorithm (decremental algorithm 2). Decremental algorithm 2 comprises decremental calculation of components S_(k+1) or x _(k+1), SS_(k+1), SX_(k+1), CS_(k+1), QS_(k+1) and QX_(k+1), and a kurtosis γ_(k+1) ⁽²⁾ may be calculated based on components SX_(k+1) and QX_(k+1) once they are calculated. Equation 407 may be used for directly decrementally calculating component S_(k+1) if component S_(k) is available. Equation 408 may be used for directly decrementally calculating component x _(k+1) if component x _(k) is available. Equation 422 is a traditional equation for calculating component SS_(k) in the computation subset X. Equation 423 is a traditional equation for calculating component SS_(k+1) in the modified computation subset X′. Equation 424 may be used for directly decrementally calculating component SS_(k+1) in the modified computation subset X′ if component SS_(k) is available. Equation 425 is a traditional equation for calculating component SX_(k) in the computation subset X. Equation 426 is a traditional equation for calculating component SX_(k+1) in the modified computation subset X′. Equations 427 may be used for decrementally calculating component SX_(k+1) in the modified computation subset X′ if components S_(k) or x _(k) and S_(k+1) or x ₊₁ and SX_(k) are available. Equations 427 comprise multiple equations but only one of them is needed depending on if a sum or a mean or both are available. Equation 428 is a traditional equation for calculating component CS_(k) in the computation subset X. Equation 429 is a traditional equation for calculating component CS_(k+1) in the modified computation subset X′. Equation 430 may be used for directly decrementally calculating component CS_(k+1) in the modified computation subset X′ if component CS_(k) is available. Equation 431 is a traditional equation for calculating component QS_(k) in the computation subset X. Equation 432 is a traditional equation for calculating component QS_(k+1) in the modified computation subset X′. Equation 433 may be used for directly decrementally calculating component QS_(k+1) in the modified computation subset X′ if component QS_(k) is available. Equation 434 is a traditional equation for calculating component QX_(k) in the computation subset X. Equation 435 is a traditional equation for calculating component QX_(k+1) in the modified computation subset X′. Equations 436 may be used for decrementally calculating component QX_(k+1) in the modified computation subset X′ if components QS_(k+1), CS_(k+1), SS_(k+1) and S_(k+1) or x _(k+1) are available. Equations 436 comprise multiple equations but only one of them is needed depending on if a sum or a mean or both are available. Equation 437 may be used for calculating the kurtosis γ_(k+1) ⁽²⁾ for the modified computation subset X′ based on components QX_(k+1) and SX_(k+1) once they are calculated.

FIG. 4E illustrates the third example decremental kurtosis calculation algorithm (decremental algorithm 3). Decremental algorithm 3 comprises decremental calculation of components S_(k+1) or x _(k+1), SS_(k+1), SX_(k+1), CS_(k+1), QS_(k+1) and QX_(k+1), and a kurtosis γ_(k+1) ⁽²⁾ may be directly calculated based on components QX_(k+1) and SX_(k+1) once they are calculated. Equation 407 may be used for directly decrementally calculating component S_(k+1) if component S_(k) is available. Equation 408 may be used for directly decrementally calculating component x _(k+1) if component x _(k) is available. Equation 438 is a traditional equation for calculating component SS_(k) in the computation subset X. Equation 439 is a traditional equation for calculating component SS_(k+1) in the modified computation subset X′. Equation 440 may be used for directly decrementally calculating component SS_(k+1) in the modified computation subset X′ if component SS_(k) is available. Equation 441 is a traditional equation for calculating component SX_(k) in the computation subset X. Equation 442 is a traditional equation for calculating component SX_(k+1) in the modified computation subset X′. Equations 443 may be used for indirectly decrementally calculating SX_(k+1) in the modified computation subset X′ if components SS_(k+1) and S_(k+1) or x _(k+1) are available. Equations 443 comprise multiple equations but only one of them is needed depending on if a sum S_(k+1) or a mean x _(k+1) or both are available. Equation 444 is a traditional equation for calculating component CS_(k) in the computation subset X. Equation 445 is a traditional equation for calculating component CS_(k+1) in the modified computation subset X′. Equations 446 may be used for directly decrementally calculating component CS_(k+1) in the modified computation subset X′ if component CS_(k) is available. Equation 447 is a traditional equation for calculating component QS_(k) in the computation subset X. Equation 448 is a traditional equation for calculating component QS_(k+1) in the modified computation subset X′. Equations 449 may be used for decrementally calculating component QS_(k+1) in the modified computation subset X′ if component QS_(k) is available. Equation 450 is a traditional equation for calculating component QX_(k) in the computation subset X. Equation 451 is a traditional equation for calculating component QX_(k+1) in the modified computation subset X′. Equations 452 may be used for indirectly decrementally calculating component QX_(k+1) in the modified computation subset X′ if components QS_(k+1), CS_(k+1), SS_(k+1) and S_(k+1) or x _(k+1) are available. Equations 452 comprise multiple equations but only one of them is needed depending on if a sum S_(k+1) or a mean x _(k+1) or both are available. Equation 453 may be used for calculating the kurtosis γ_(k+1) ⁽²⁾ for the modified computation subset X′ based on components QX_(k+1) and SX_(k+1) once they are calculated.

To demonstrate decremental kurtosis calculation algorithms and their comparison against traditional algorithms, three examples are given below. Three computation subsets of data elements are used. For traditional algorithms, the calculations for all three computation subsets are exactly the same. For decremental algorithms, initialization of one or more components is performed for the first computation subset, and decremental calculations are performed for the second and third computation subsets.

FIG. 5A illustrates an example of calculating a kurtosis on Big Data Set 501 using traditional algorithms. Big Data Set 501 is either a Big Data set or streamed data. Computation subset 502 includes 6 data elements in Big Data Set 501. Computation subset size 503 (n) is 6. There are a total of 2 divisions, 20 multiplications, 15 additions, and 12 subtractions when calculating the kurtosis γ₁ ⁽²⁾ on 6 data elements without any optimization.

The same equations may be used to calculate the kurtosis for computation subset 504 as shown in FIG. 5A Cont'd 1, however the computation subset size 505 is decreased to 5. The calculation includes a total of 2 divisions, 17 multiplications, 12 additions, and 10 subtractions when calculating the kurtosis γ₂ ⁽²⁾ on 5 data elements without any optimization.

The same equations may be used to calculate the kurtosis for computation subset 506 as shown in FIG. 5A Cont'd 2. The computation subset size 507 is decreased to 4. The calculation includes a total of 2 divisions, 14 multiplications, 9 additions, and 8 subtractions when calculating the kurtosis γ₃ ⁽²⁾ on 4 data elements without any optimization. Traditional algorithms for calculating kurtosis on n data elements will typically take 2 divisions, 3n+2 multiplications, 3(n−1) additions, and 2n subtractions without any optimization.

FIG. 5B illustrates an example of calculating a kurtosis using decremental algorithm 1. The calculations for computation subset 502 uses traditional equations to calculate the initial values of components x ₁, SS₁, SX₁, CS₁ and QX₁. For example, equation 402 may be used for calculating the component x ₁. Equation 409 is a traditional equation for calculating SS₁ in the computation subset X. Equation 412 is a traditional equation for calculating SX₁ in the computation subset X. Equation 415 is a traditional equation for calculating CS₁ in the computation subset X. Equation 418 is a traditional equation for calculating QX₁ in the computation subset X. Equation 403 is an equation for calculating kurtosis γ₁ ⁽²⁾ in the computation subset X. There are a total of 2 divisions, 26 multiplications, 25 additions and 6 subtractions when calculating the kurtosis. In practice, using traditional algorithms to calculate those components will not happen, because decremental kurtosis calculation would start from a non-empty computation subset where those components have already been calculated. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the decremental kurtosis calculation algorithm. The kurtosis of computation subset 502 is then calculated based on those components.

However, for subset 504, the components of the kurtosis for computation subset 504 may be decrementally calculated from the components of the kurtosis for computation subset 502. The computation subset size 505 is decreased to 5. For example, equation 408 may be used for decrementally calculating the component x ₂ based on component x ₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 411 may be used for decrementally calculating the component SS₂ based on component SS₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 414 may be used for decrementally calculating the component SX₂ based on components SS₂ and x ₂. Equation 417 may be used for decrementally calculating the component CS₂ based on CS₁ and removing any contribution of the removed data element x_(r) mathematically. Equation 420 may be used for decrementally calculating the component QX₂ based on components QX₁, CS₂, SS₂, x ₁ and x ₂ and removing any contribution of the removed data element x_(r) mathematically. Equation 421 may be used for calculating the kurtosis γ₂ ⁽²⁾ based on components QX₂ and SX₂. There are a total of 2 divisions, 17 multiplications, 4 additions and 12 subtractions when calculating the kurtosis.

The same equations may also be used for decrementally calculating the components of kurtosis for computation subset 506 from the components of kurtosis for computation subset 504. The computation subset size 507 is decreased to 4. For example, equation 408 may be used for decrementally calculating the component x ₃ based on component x ₂ previously calculated for computation subset 504 and removing any contribution of the removed data element x_(r) mathematically. Equation 411 may be used for decrementally calculating the component SS₃ based on component SS₂ previously calculated for computation subset 504 and removing any contribution of the removed data element x_(r) mathematically. Equation 414 may be used for decrementally calculating the component SX₃ based on components SS₃ and x ₃. Equation 417 may be used for decrementally calculating the component CS₃ based on CS₂ and removing any contribution of the removed data element x_(r) mathematically. Equation 420 may be used for decrementally calculating the component QX₃ based on components QX₂, CS₃, SS₃, x ₂ and x ₃ and removing any contribution of the removed data element x_(r) mathematically. Equation 421 may be used for calculating the kurtosis γ₃ ⁽²⁾ based on components QX₃ and SX₃. There are still a total of 2 divisions, 17 multiplications, 4 additions and 12 subtractions when decrementally calculating the components of kurtosis. As such, starting from computation subset 504, the number of operations used when decrementally calculating the kurtosis is (potentially substantially) less than when using traditional equations.

FIG. 5C illustrates an example of calculating a kurtosis using decremental algorithm 2. A mean instead of a sum is used in this example. The calculations for computation subset 502 uses traditional equations to calculate the initial values of components x ₁, SS₁, SX₁, CS₁, QS₁ and QX₁ and then use SX₁ and QX₁ to calculate the kurtosis γ₁ ⁽²⁾. For example, equation 402 may be used for calculating component x ₁. Equation 422 may be used for calculating component SS₁. Equation 425 may be used for calculating component SX₁. Equation 428 may be used for calculating component CS₁. Equation 431 may be used for calculating component QS₁. Equation 434 may be used for calculating component QX₁. The kurtosis γ₁ ⁽²⁾ of computation subset 502 is then calculated based on components QX₁ and SX₁ through equation 437. There are a total of 2 divisions, 32 multiplications, 30 additions and 6 subtractions when calculating the kurtosis without any optimizations. In practice, for decremental kurtosis calculation, using traditional algorithms to do an initial component calculation will not happen, because decremental kurtosis calculation would start from a non-empty computation subset where those components have already been calculated. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the decremental kurtosis calculation algorithm.

However, for subset 504, the components of the kurtosis for computation subset 504 may be decrementally calculated from the components of the kurtosis for computation subset 502. For example, equation 408 may be used for decrementally calculating the component x ₂ based on x ₁ previously calculated for computation subset 502 and mathematically removing any contribution of the removed data element x_(r). Equation 424 may be used for decrementally calculating the component SS₂ based on SS₁ previously calculated for computation subset 502 and mathematically removing any contribution of the removed data element x_(r). Equation 427 may be used for decrementally calculating the component SX₂ based on components x ₁, x ₂ and SX₁ and removing any contribution of the removed data element x_(r) mathematically. Equation 430 may be used for decrementally calculating CS₂ based on CS₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 433 may be used for decrementally calculating the component QS₂ based on QS₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 436 may be used for decrementally calculating the component QX₂ based on QS₂, SS₂, CS₂ and x ₂ previously calculated for computation subset 502. Equation 437 may then be used for calculating the kurtosis γ₂ ⁽²⁾ based on components QX₂ and SX₂. There are a total of 2 divisions, 14 multiplications, 1 addition and 11 subtractions when calculating the kurtosis.

The same equations may also be used for decrementally calculating the components of kurtosis for computation subset 506 from the components of kurtosis for computation subset 504. The computation subset size 507 is decreased to 4. There are still a total of 2 divisions, 14 multiplications, 1 addition and 11 subtractions when decrementally calculating the components of kurtosis γ₃ ⁽²⁾. As such, starting from computation subset 504, the number of operations used when decrementally calculating the kurtosis is (potentially substantially) less than when using traditional equations.

FIG. 5D illustrates an example of calculating a kurtosis using decremental algorithm 3. The calculations for computation subset 502 uses traditional equations to calculate the initial values of components x ₁, SS_(k), SX₁, CS₁, QS₁ and QX₁ and then use SX₁ and QX₁ to calculate the kurtosis γ₁ ⁽²⁾. For example, equation 402 may be used for calculating component x ₁. Equation 438 may be used for calculating SS₁. Equation 441 may be used for calculating SX₁. Equation 444 may be used for calculating CS₁. Equation 447 may be used for calculating QS₁. Equation 450 may be used for calculating QX₁. Equation 453 may then be used for calculating the kurtosis γ₁ ⁽²⁾ of computation subset 502 based on components QX₁ and SX₁. There are a total of 2 divisions, 32 multiplications, 30 additions and 6 subtractions when calculating the kurtosis. In practice, for decremental kurtosis calculation, using traditional algorithms to do an initial component calculation will not happen, because decremental kurtosis calculation would start from a non-empty computation subset where those components have already been calculated. Using traditional algorithms to calculate those components on this non-empty computation subset here is only for the purpose of illustrating the decremental kurtosis calculation algorithm.

However, for subset 504, the components of the kurtosis for computation subset 504 may be decrementally calculated from the components of the kurtosis for computation subset 502. For example, equation 408 may be used for decrementally calculating the component x ₂ based on x ₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 440 may be used for decrementally calculating the component SS₂ based on SS₁ previously calculated for computation subset 502 and removing any contribution of the removed data element x_(r) mathematically. Equation 443 may be used for decrementally calculating the component SX₂ based on components SS₂ and x ₂ Equation 446 may be used for decrementally calculating the component CS₂ based on components CS₁ and removing any contribution of the removed data element x_(r) mathematically. Equation 449 may be used for decrementally calculating the component QS₂ based on components QS₁ and removing any contribution of the removed data element x_(r) mathematically. Equation 452 may be used for decrementally calculating the component QX₂ based on components QS₂, CS₂, SS₂ and x ₂. Equation 453 may then be used for calculating the kurtosis γ₂ ⁽²⁾ based on components QX₂ and SX₂. There are a total of 2 divisions, 15 multiplications, 1 addition and 8 subtractions when calculating the kurtosis.

The same equations may also be used for decrementally calculating the components of kurtosis for computation subset 506 from the components of kurtosis for computation subset 504. The computation subset size 507 is decreased to 4. There are a total of 2 divisions, 15 multiplications, 1 addition and 8 subtractions when decrementally calculating the components of kurtosis γ₃ ⁽²⁾. As such, starting from computation subset 504, the number of operations used when decrementally calculating the kurtosis is (potentially substantially) less than when using traditional equations.

In the three examples above, a mean is used for the decremental kurtosis calculation. If a sum instead of a mean is used, kurtosis may also be decrementally calculated though the numbers of operations are different.

FIG. 6 illustrates computational loads for traditional kurtosis algorithm and decremental kurtosis algorithms for n=4 for computation subset 506. As depicted, there are fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the decremental algorithms compared to traditional algorithms.

FIG. 7 illustrates computational loads for traditional kurtosis algorithm and decremental kurtosis algorithms for n=1,000,000 for any computation subset other than the first computation subset. Again, as depicted, there are fewer multiplication operations, fewer addition operations, and fewer subtraction operations using any one of the decremental algorithms compared to traditional algorithms.

The present invention may be embodied in other specific forms without departing from its spirit or essential characteristics. The described embodiments are to be considered in all respects only as illustrative and not restrictive. The scope of the invention is, therefore, indicated by the appended claims rather than by the foregoing description. All changes which come within the meaning and range of equivalency of the claims are to be embraced within their scope. 

What is claimed:
 1. A computing-system-implemented method for calculating a kurtosis for a modified computation subset, the method comprising: initializing, by a computing-device-based computing system, a computation subset size counter, at least one of a sum and a mean, and one or more other components of a kurtosis for a pre-modified computation subset of a data stream or a data set on at least one of one or more storage media, wherein the pre-modified computation subset contains a specified number, n (n≥6), of data elements from the data stream or the data set; accessing or receiving, by the computing-device-based computing system, a data element to be removed from the pre-modified computation subset; modifying, by the computing-device-based computing system, the pre-modified computation subset by: removing the to-be-removed data element from the pre-modified computation subset; and modifying the computation subset size counter by decreasing its value by 1; decrementally deriving, by the computing-device-based computing system, at least one of a sum and a mean for the modified computation subset; directly decrementally deriving, by the computing-device-based computing system, one or more components of a kurtosis other than a sum and a mean for the modified computation subset based at least in part on the one or more components of the kurtosis other than a sum and a mean initialized or derived for the pre-modified computation subset, wherein the directly decrementally deriving includes: accessing the one or more components of the kurtosis other than a sum and a mean for the pre-modified computation subset without accessing all data elements in the modified computation subset thereby saving computing resources and reducing the computing-device-based computing system's power consumption; and removing any contribution of the data element removed from the pre-modified computation subset from each of the accessed components mathematically wherein not all data elements in the modified computation subset are used in the computation to reduce number of operations performed by the computing-device-based computing system thereby increasing calculation efficiency; and generating, by the computing-device-based computing system, a kurtosis for the modified computation subset based on one or more of the decrementally derived components.
 2. The computing-system-implemented method of claim 1, wherein the generating a kurtosis for the modified computation subset further comprises indirectly decrementally deriving, by the computing-device-based computing system, one or more components of the kurtosis for the modified computation subset, wherein the indirectly decrementally deriving the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 3. The computing-system-implemented method of claim 1, wherein accessing or receiving a data element to be removed from the computation subset includes accessing or receiving a plurality of z (z>1) data elements to be removed from the pre-modified computation subset, and wherein the method further comprises decrementally performing, for each of the respective z data elements, the modifying the pre-modified computation subset, the decrementally deriving at least one of a sum and a mean for the modified computation subset, the directly decrementally deriving the one or more components of a kurtosis other than a sum and a mean for the modified computation subset, and the generating a kurtosis for the modified computation subset.
 4. The computing-system-implemented method of claim 3, wherein the generating a kurtosis for the modified computation subset comprises generating a kurtosis for the modified computation subset only when the kurtosis is accessed.
 5. The computing-system-implemented method of claim 4, wherein the generating a kurtosis for the modified computation subset further comprises indirectly decrementally deriving, by the computing-device-based computing system, one or more components of the kurtosis for the modified computation subset, wherein the indirectly decrementally deriving the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 6. The computing-system-implemented method of claim 1, wherein accessing or receiving a data element to be removed from the computation subset includes accessing or receiving a plurality of z (z>1) data elements to be removed from the pre-modified computation subset, and wherein the method further comprises decrementally performing, for each of the respective z data elements, the modifying the pre-modified computation subset, the decrementally deriving at least one of a sum and a mean for the modified computation subset, and the directly decrementally deriving the one or more components for the modified computation subset.
 7. The computing-system-implemented method of claim 1, wherein the accessing or receiving a data element to be removed from the pre-modified computation subset further comprises removing the accessed or received data element from a data buffer on at least one of the one or more storage media.
 8. A computing system, the computing system comprising: one or more computing devices; each computing device comprising one or more processors; one or more storage media; and one or more calculation modules that, when executed by at least one of the one or more computing devices, determine a kurtosis for a modified computation subset, the one or more calculation modules configured to: a. initialize a computation subset size counter, at least one of a sum and a mean, and one or more other components of a kurtosis for a pre-modified computation subset of a data stream or a data set on at least one of the one or more storage media, wherein the pre-modified computation subset contains a specified number, n (n≥6), of data elements from the data stream or the data set; b. access or receive a data element to be removed from the pre-modified computation subset; c. modify the pre-modified computation subset by removing the to-be-removed data element from the pre-modified computation subset and modifying the computation subset size counter by decreasing its value by 1; d. decrementally calculate at least one of a sum and a mean for the modified computation subset; e. directly decrementally calculate one or more components of a kurtosis other than a sum and a mean for the modified computation subset based at least in part on the one or more components of the kurtosis other than a sum and a mean initialized or calculated for the pre-modified computation subset, wherein decremental calculation of the one or more components includes to remove any contribution of the removed data element from each of the one or more components of the kurtosis initialized or calculated for the pre-modified computation subset mathematically without accessing and using all data elements in the modified computation subset thereby reducing data access latency, reducing number of operations performed by the computing system, and reducing the computing system's power consumption; and f. generate a kurtosis for the modified computation subset based on one or more of the decrementally calculated components.
 9. The computing system of claim 8, wherein the generating a kurtosis for the modified computation subset further comprises indirectly decrementally calculate one or more components of a kurtosis for the modified computation subset, wherein indirectly decrementally calculating one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 10. The computing system of claim 8, wherein the one or more calculation modules, when executed by at least one of the one or more computing devices, perform, c, d, e, and f each time b is performed.
 11. The computing system of claim 10, wherein the performing f comprises generating a kurtosis for the modified computation subset only when the kurtosis is accessed.
 12. The computing system of claim 11, wherein the generating a kurtosis for the modified computation subset further comprises indirectly decrementally calculate one or more components of the kurtosis for the modified computation subset, wherein indirectly decrementally calculating one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 13. The computing system of claim 8, wherein the one or more calculation modules, when executed by at least one of the one or more computing devices, perform c, d, and e each time b is performed.
 14. The computing system of claim 8, wherein the accessing or receiving a data element to be removed from the pre-modified computation subset further comprises removing the accessed or received data element from a data buffer on at least one of the one or more storage media.
 15. A computing system program product for use at a configured computing system which comprises one or more computing devices and one or more storage media, each computing device comprising one or more processors, the computing system program product comprising one or more non-transitory computing-device-readable storage media having stored thereon computing-device-executable instructions that, when executed by at least one of the one or more computing devices in the configured computing system, cause the configured computing system to perform a method for generating a kurtosis for data elements in a modified computation subset, the method including steps to: initialize, by the configured computing system, a computation subset size counter, at least one of a sum and a mean, and one or more other components of a kurtosis for a pre-modified computation subset of a data stream or a data set on at least one of the one or more storage media, wherein the pre-modified computation subset contains a specified number, n (n≥6), of data elements from the data stream or the data set; access or receive, by the configured computing system, a data element to be removed from the computation subset; modify, by the configured computing system, the pre-modified computation subset via removing the to-be-removed data element from the pre-modified computation subset and modifying the computation subset size counter by decreasing its value by 1; decrementally calculate, by the configured computing system, at least one of a sum and a mean for the modified computation subset; directly decrementally calculate, by the configured computing system, one or more components of a kurtosis other than a sum and a mean for the modified computation subset based at least in part on the one or more components of the kurtosis other than a sum and a mean initialized or calculated for the pre-modified computation subset, including to: access the one or more components of the kurtosis other than a sum and a mean for the pre-modified computation subset without accessing all data elements in the modified computation subset thereby saving computing resources and reducing the computing-device-based computing system's power consumption; and remove any contribution of the data element removed from the pre-modified computation subset from each of the accessed components mathematically wherein not all data elements in the modified computation subset are used in the computation to reduce number of operations performed by the computing-device-based computing system thereby increasing calculation efficiency; and generate, by the configured computing system, a kurtosis for the modified computation subset based on one or more of the decrementally calculated components.
 16. The computing system program product of claim 15, wherein the generating a kurtosis further comprises indirectly decrementally calculating, by the configured computing system, one or more components of the kurtosis for the modified computation subset, wherein indirectly decrementally calculating the one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 17. The computing system program product of claim 15, wherein the computing-device-executable instructions that, when executed, further cause the configured computing system to modify the pre-modified computation subset, to decrementally calculate at least one of a sum and a mean, to directly decrementally calculate the one or more components other than a sum and a mean, and to generate a kurtosis for the modified computation subset for each additional data element accessed or received.
 18. The computing system program product of claim 17, wherein the computing-device-executable instructions that, when executed, cause the configured computing system to generate a kurtosis comprise computing-device-executable instructions that, when executed, cause the configured computing system to generate a kurtosis for the modified computation subset only when the kurtosis is accessed.
 19. The computing system program product of claim 18, wherein the generating a kurtosis for the modified computation subset further comprises indirectly decrementally calculate one or more components of the kurtosis for the modified computation subset, wherein indirectly decrementally calculating one or more components includes individually calculating each respective one of the one or more components based on one or more components other than the respective one component.
 20. The computing system program product of claim 15, wherein the computing-device-executable instructions that, when executed, further cause the configured computing system to modify the pre-modified computation subset, to decrementally calculate at least one of a sum and a mean, and to directly decrementally calculate the one or more components other than a sum and a mean for each additional data element accessed or received. 